Abstract and Applied AnalysisVolume 2011 (2011), Article ID 574756, 8 pagesdoi:10.1155/2011/574756

Research Article

Modular Locally Constant Mappings in Vector Ultrametric Spaces

Kamal Fallahi¹ and Kourosh Nourouzi1,2

¹Department of Mathematics, K. N. Toosi University of Technology, P.O. Box 16315-1618, Tehran 1541849611, Iran²School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran

Received 16 November 2010; Revised 15 February 2011; Accepted 23 February 2011

Academic Editor: Norimichi Hirano

Copyright © 2011 Kamal Fallahi and Kourosh Nourouzi. Kamal Fallahi and . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We give some sufficient conditions for mappings defined on vector ultrametric spaces to be modular locally constant.

1. Introduction and Preliminaries

A metric space in which the triangle inequality is replaced by is called an ultrametric space. Generalized ultrametric spaces were given in [1, 2] via partially ordered sets and some applications of them appeared in logic programming [3], computational logic [4], and quantitative domain theory [5].

In [6], the notion of a metric locally constant function on an ultrametric space was given in order to investigate certain groups of isometries and describe various Galois groups over local fields. Locally constant functions also appear in contexts such as higher ramification groups of finite extensions of , and the Fontaine ring . Also, metric locally constant functions were studied in [7, 8]. On the other hand, vector ultrametric spaces are given in [9] as vectorial generalizations of ultrametrics. Hence, locally constant functions, in modular sense, can play the same role in vector ultrametric spaces as they do in usual ultrametric spaces.

In this paper, we introduce modular locally constant mappings in vector ultrametric spaces. Some sufficient conditions are given for mappings defined on vector ultrametric spaces to be modular locally constant.

We first present some basic notions.

Recall that a modular on a real linear space is a real valued functional on satisfying the conditions: (1) if and only if , (2), (3), for all and , .

Then, the linear subspace of is called a modular space.

A sequence in is called -convergent (briefly, convergent) to if as , and is called Cauchy sequence if as . By a -closed (briefly, closed) set in we mean a set which contains the limit of each of its convergent sequences. Then, is a complete modular space if every Cauchy sequence in is convergent to a point of . We refer to [10, 11] for more details.

A cone in a complete modular space is a nonempty set such that

(i) is -closed, and ;

(ii) , , ;

(iii) , where .

Let be the partial order on induced by the cone , that is, whenever . The cone is called normal if The cone is said to be unital if there exists a vector with modular 1 such that

Example 1.1. Consider the real vector space consisting of all real-valued continuous functions on equipped with the modular defined by It is not difficult to see that is a complete modular space and is a normal cone in .

Example 1.2. The vector space consisting of all continuously differentiable real-valued functions on equipped with the modular defined by constitutes a complete modular space. The subset is a unital cone in with unit 1. The cone is not normal since, for example, , for does not imply that .

Throughout this note, we suppose that is a cone in complete modular space , and is the partial order induced by .

Definition 1.3. A vector ultrametric on a nonempty set is a mapping satisfying the conditions:(CUM1) for all and if and only if ; (CUM2) for all ; (CUM3) If and , then , for any , and . Then the triple is called a vector ultrametric space. If is unital and normal, then is called a unital-normal vector ultrametric space.

For unital-normal vector ultrametric space , since from (CUM3) we get and therefore Let be a unital-normal vector ultrametric space. If and , the ball centered at with radius is defined as The unital-normal vector ultrametric space is called spherically complete if every chain of balls (with respect to inclusion) has a nonempty intersection.

The following lemma may be easily obtained.

Lemma 1.4. Let be a unital-normal vector ultrametric space. (1)If , and , then . (2)If , , then either or .

Definition 1.5. Let be a unital-normal vector ultrametric space. A mapping is said to be modular locally constant provided that for any and any one has .

2. Main Theorem

Theorem 2.1. Let be a spherically complete unital-normal vector ultrametric space and be a mapping such that for every , either or Then there exists a subset of such that and the mapping is modular locally constant.

Proof. Let where . Consider the partial order on defined by where . If is any chain in , then the spherical completeness of implies that the intersection of elements of is nonempty. Suppose that (2.1) holds. Let and . Obviously , so . For any , we have So for every , ; that is, is an upper bound in for the family . By Zorn's lemma, there exists a maximal element in , say . If , , and we get Then Since , we have by Lemma 1.4. But , so . Now we show that for every . It is clear that , for all . Suppose for some . We have , and which is a contradiction. Thus is modular locally constant on . Suppose that (2.2) holds. As above, let and . Obviously , so . For any , we have Thus So, for every , ; that is, is an upper bound for the family . Again, by Zorn's lemma there exists a maximal element in , say . For any , we have This implies that , and Lemma 1.4 gives . Since , so .If , then on and this yields the result. If , we show that for every . Since for any , let us suppose that for some , . So and thus . But implies that , but and hence which contradicts the maximality of . This completes the proof.

In the following, we assume that is a spherically complete unital-normal vector ultrametric space.

Corollary 2.2. Let be a mapping such that for all , Then there exists a subset of such that and the mapping defined in (2.3) is modular locally constant.

Proof. Since for all , we get for all . Now, if then which is a contradiction. Thus , and so Therefore Now, Theorem 2.1 completes the proof.

Corollary 2.3. Let be a mapping such that for all , Then there exists a subset of such that and the mapping defined in (2.3) is modular locally constant.

Proof. We have for all . Again, Theorem 2.1, completes the proof.

Acknowledgments

The authors would like to thank the referee for his/her valuable comments on this paper. The second author’s research was in part supported by a grant from IPM (No. 89470128).

References

S. Priess-Crampe and P. Ribenboim, “Generalized ultrametric spaces. I,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 66, pp. 55–73, 1996.

S. Priess-Crampe and P. Ribenboim, “Generalized ultrametric spaces. II,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 67, pp. 19–31, 1997.

S. Priess-Crampe and P. Ribenboim, “Ultrametric spaces and logic programming,” Journal of Logic Programming, vol. 42, no. 2, pp. 59–70, 2000.

A. K. Seda and P. Hitzler, “Generalized ultrametrics, domains and an application to computational logic,” Irish Mathematical Society Bulletin, no. 41, pp. 31–43, 1998.

M. Krötzsch, “Generalized ultrametric spaces in quantitative domain theory,” Theoretical Computer Science, vol. 368, no. 1-2, pp. 30–49, 2006.

M. Vâjâitu and A. Zaharescu, “Groups of isometries on ultrametric spaces,” Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie, vol. 44(92), no. 2, pp. 183–191, 2001.

L. Gajić, “Metric locally constant function on some subset of ultrametric space,” Novi Sad Journal of Mathematics, vol. 35, no. 1, pp. 123–125, 2005.

M. Vâjâitu and A. Zaharescu, “Metric locally constant functions,” Acta et Commentationes Universitatis Tartuensis de Mathematica, no. 6, pp. 29–36, 2002.

K. Nourouzi, “Vector ultrametric spaces and a fixed point theorem for correspondences,” submitted to Numerical Functional Analysis and Optimization.

W. M. Kozlowski, Modular Function Spaces, vol. 122 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1988.

J. Musielak, Orlicz Spaces and Modular Spaces, vol. 1034 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1983.